Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 2 / c / i / Solution

The chain alternates between the classes $\{1,2\}$ and $\{3,4\}$. It is irreducible and has period two, since no odd return is possible and each state has a positive two-step return probability. Matrix multiplication givesThe even-time sampled chain stays in its initial class and moves between its two states with probability $4/9$ at each sampled step.